“The shop was closed; a light still burned upstairs.” The semicolon holds two complete statements together without adding a word that explains their relationship. That can be useful, but the mark does not turn an unfinished phrase into a complete thought.
Test each side on its own
In the example, “The shop was closed” and “A light still burned upstairs” can each stand as a sentence. Purdue OWL’s discussion of commas and semicolons explains the use of a semicolon between independent clauses in this kind of construction.
Now try “The shop was closed; despite the light upstairs.” The second part does not stand independently in the same way. You need to revise the structure rather than assuming that a stronger-looking punctuation mark solves it.
Ask why the statements belong together
The first example invites a connection between closure below and activity above. If the next clause concerned an unrelated bus timetable, the grammatical construction might still be possible, but the relationship could become harder to follow.
Write the passage once with a full stop and once with a semicolon. Read both in context. The choice concerns the connection and rhythm you want here, not a ranking in which one mark is more sophisticated.
Keep the check local during revision
Search a draft for semicolons and inspect each occurrence separately. This exercise addresses their use between clauses; semicolons can also have other roles, such as separating complicated list items.
Our note on reading a paragraph aloud offers a useful second pass. First confirm the structure, then listen to the movement. Keep a mark because it helps the reader follow the thought, not because the sentence looks more literary with it.